(15,14) “And Shmuel said: What is this sound of sheep in my
ears, and the sound of cattle that I hear? And Shaul said: They have
brought them from the Amalekites; for the people spared the best of the
sheep and of the oxen to sacrifice to Hashem, your G-d; and the rest we
have utterly destroyed.”

We can explain this exchange according to the Midrash which
asked: Granted the Amalekites were killed because they had sinned, but
what was the animals' sin? It answered that the Amalekites were
sorcerers and could change themselves into animals in order to save
themselves. Therefore, Hashem commanded them to kill the animals as
well.

Now, it says in Sefer Chasidim that those who change
themselves into an animal are able to change their entire body except
for their eyes, which remain the same.
And in the gemora Bechoros it says that one of the blemishes which
disqualify an animal as a sacrifice is if the eyes are similar to the
eyes of a human.

Now we understand the exchange between Shmuel and Shaul:
“Shmuel said: What is the sound of sheep which is in my ears” - you
were supposed to also kill the sheep in case some of them are really
humans. To this Shaul responded that “the people spared the best of the
sheep to sacrifice to Hashem”. And since they took them only for the
purpose of sacrificing, presumably they checked them for blemishes,
including a check that their eyes are not similar to the eyes of
humans. Therefore, none of them are Amalekites who changed themselves
into animals, and so you have no cause to complain about our leaving
these animals alive.

Why did Hashem reject Shaul from
being king over Yisrael?

(15,26) “And Shmuel said to Shaul: I will not return with you,
for you have rejected the word of Hashem, and Hashem has rejected you
from being a king over Yisrael.”

We can explain how Hashem’s rejection of Shaul from being king
was a direct result of his not killng the animals of the Amalekites
according to the well known explanation of why the first king of
Yisrael was chosen from the tribe of Binyamin, who was the youngest of
the tribes. The answer that is given is that when Ya'akov met up with
his brother Eisav in parshas Vayishlach, all the tribes bowed down to
Eisav apart from Binyamin who had not yet been born.

According to this we can explain that Shmuel was saying to
Shaul as follows: Hashem has rejected you from being king over Yisrael
because you kept some of the animals of the Amalekites instead of
destroying
them. And even though you wish to justify your actions by saying that
they
were saved in order to sacrifice them to Hashem, you should still be
guilty of
the sin of 'maris ayin', because people might think that you
took the animals for yourself. Therefore, the only way that your
defence of your actions can be justified, is if you hold that 'maris
ayin' is not a sin.

But the consequence of holding this opinion is that Hashem
has rejected you from being king. Because why did Hashem choose you,
the youngest of all the tribes, to be the king? Only
because all the other brothers bowed down to Eisav. And even though the
truth is that did not bow
down to Eisav but rather to Hashem, they were nevertheless guilty of
the sin of 'maris ayin'. But according to your opinion this is not a
sin, and so the
kingship was never fitting for you, but for Yehudah, "because Yehudah
prevailed over his brothers, and the one appointed as prince was to be
from
him" (Divrei Hayomim 5:2).